Rice's formula

In probability theory, Rice's formula counts the average number of times an ergodic stationary process X(t) per unit time crosses a fixed level u.

Rice in 1944,[4] having previously been discussed in his 1936 note entitled "Singing Transmission Lines.

"[5][6] Write Du for the number of times the ergodic stationary stochastic process x(t) takes the value u in a unit of time (i.e. t ∈ [0,1]).

Then Rice's formula states that where p(x,x') is the joint probability density of the x(t) and its mean-square derivative x'(t).

[7] If the process x(t) is a Gaussian process and u = 0 then the formula simplifies significantly to give[7][8] where ρ'' is the second derivative of the normalised autocorrelation of x(t) at 0.